Abstract
This article studies the sensitivity of Granger causality to the addition of noise, the introduction of subsampling, and the application of causal invertible filters to weakly stationary processes. Using canonical spectral factors and Wold decompositions, we give general conditions under which additive noise or filtering distorts Granger‐causal properties by inducing (spurious) Granger causality, as well as conditions under which it does not. For the errors‐in‐variables case, we give a continuity result, which implies that: a ‘small’ noise‐to‐signal ratio entails ‘small’ distortions in Granger causality. On filtering, we give general necessary and sufficient conditions under which ‘spurious’ causal relations between (vector) time series are not induced by linear transformations of the variables involved. This also yields transformations (or filters) which can eliminate Granger causality from one vector to another one. In a number of cases, we clarify results in the existing literature, with a number of calculations streamlining some existing approaches.
Highlights
Granger causality is one of the most important concepts for the analysis of the structure of multivariate time series
We study the effect of additive noise on Granger causality in the context of a general weakly stationary multivariate model, especially in view of finding when spurious causality could appear, and when properties of non-causality are unaffected by measurement errors
We review some classical characterizations of Granger causality which will be useful for studying the effect of errors-in-variables
Summary
Granger causality is one of the most important concepts for the analysis of the structure of multivariate time series. General characterizations of non-causality are typically little affected when common forms of forms of non-stationarity – such deterministic time trends and integration) – are allowed; see, for example, Dufour and Renault (1998) and Dufour et al (2006) It is clear from the above definitions that Granger causality depends on the vector X considered and on the way X is split into subvectors XA and XB. These include : a general lower bound on the conditional variance of the sum of two processes, and some general relations between Granger causality and instantaneous causality.
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